
where \(\mathcal {W}_l \in \mathbb {C}^{\tau _1\times \cdots \tau _d\times M}\)  is the additive white Gaussian noise represented in the tensor space. Therefore, in (REF ), each active device transmits a sequence that is associated with a rank-one tensor of order \(d\) , while \(\mathcal {Y}_l^0\)  is the noise-free tensor of order
\(d +1\)  and has dimensions \(\tau _1, \cdots ,\tau _d ,M\) , respectively. Its tensor rank is at most \(K_a\) , since a tensor is said to be rank-\(K_a\)  whenever \(K_a\)  is the smallest integer such that the tensor can be written as a sum of \(K_a\)  rank-1 tensors [1]}.
